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Linear Recurrence

  1. Dec 31, 2004 #1
    Consider the recurrence x_k+2 = ax_k+1 + bx_k + c where c may not be zero.

    If a + b is not equal to 1 show that p can be found such that, if we set y_k = x_k + p, then y_k+2 = ay_k+1 + by_k. [Hence, the sequence x_k can be found provided y_k can be found]

    First of all, sorry about the messiness, I don't know how to use LaTeX. Now, this is the question exactly as it is from the question sheet. My problem is, I don't understand the question. And its kind of really hard to start the question without understanding it :mad: . My biggest concern is, what the heck is p and where does it come from? The way I read it, p is just -c.

    Thx in advance for any help.
  2. jcsd
  3. Dec 31, 2004 #2


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    Homework Helper

    1.You should learn "tex".
    2.Hypothetis:[tex] x_{k+2}=ax_{k+1}+bx_{k}+c [/tex] (1)
    [tex] y_{k}=x_{k}+p [/tex] (2)
    [tex] y_{k+2}=ay_{k+1}+by_{k} [/tex] (3)
    3.Question:[tex] p=...? [/tex]

    4.From (2) u have:
    [tex] y_{k+2}=x_{k+2}+p [/tex] (4)
    Combining (1) and (4),u get:
    [tex] y_{k+2}=ax_{k+1}+bx_{k}+c+p[/tex] (5)
    Equate (5) with (3),make use of (2) and extract 'p':

    Answer:[tex] p=\frac{c}{a+b-1} [/tex]

  4. Jan 1, 2005 #3
    Ah, it all makes sense. Can't believe I never saw that. Thx alot!
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